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Deformations of the Veronese embedding and Finsler 2-spheres of constant curvature

Christian Lange, Thomas Mettler

Abstract

We establish a one-to-one correspondence between Finsler structures on the $2$-sphere with constant curvature $1$ and all geodesics closed on the one hand, and Weyl connections on certain spindle orbifolds whose symmetric Ricci curvature is positive definite and all of whose geodesics are closed on the other hand. As an application of our duality result, we show that suitable holomorphic deformations of the Veronese embedding $\mathbb{CP}(a_1,a_2)\to \mathbb{CP}(a_1,(a_1+a_2)/2,a_2)$ of weighted projective spaces provide examples of Finsler $2$-spheres of constant curvature whose geodesics are all closed.

Deformations of the Veronese embedding and Finsler 2-spheres of constant curvature

Abstract

We establish a one-to-one correspondence between Finsler structures on the -sphere with constant curvature and all geodesics closed on the one hand, and Weyl connections on certain spindle orbifolds whose symmetric Ricci curvature is positive definite and all of whose geodesics are closed on the other hand. As an application of our duality result, we show that suitable holomorphic deformations of the Veronese embedding of weighted projective spaces provide examples of Finsler -spheres of constant curvature whose geodesics are all closed.

Paper Structure

This paper contains 18 sections, 13 theorems, 104 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. A duality result
  4. Construction of examples
  5. Preliminaries
  6. Background on orbifolds
  7. Besse orbifolds
  8. Finsler structures.
  9. Weyl structures and connections
  10. A Duality Theorem
  11. Construction of Examples
  12. The twistor space
  13. The Fubini--Study metric on the weighted projective line
  14. Constructing the twistor space
  15. Embedding the twistor space
  16. ...and 3 more sections

Key Result

Theorem A

There is a one-to-one correspondence between Finsler structures on $S^2$ with constant Finsler--Gauss curvature $1$ and all geodesics closed on the one hand, and positive Besse--Weyl structures on spindle orbifolds $S^2(a_1,a_2)$ with $c:=\gcd(a_1,a_2)\in \{1,2\}$, $a_1\geqslant a_2$, $2|(a_1+a_2)$ and these assignments are inverse to each other. Moreover, two such Finsler metrics are isometric i

Figures (1)

  • Figure : Isometric embeddings of $\mathbb{CP}(3,1)$ and $\mathbb{CP}(5,3)$

Theorems & Definitions (34)

  • Theorem A
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Example 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Definition 2.6
  • Lemma 2.7
  • ...and 24 more